Let${\mathbb K}$be an algebraically bounded structure, and letTbe its theory. IfTis model complete, then the theory of${\mathbb K}$endowed with a derivation, denoted by$T^{\delta }$, has a model completion. Additionally, we prove that if the theoryTis stable/NIP then the model completion of$T^{\delta }$is also stable/NIP. Similar results hold for the theory with several derivations, either commuting or non-commuting.

The following strong form of density of definable types is introduced for theoriesTadmitting a fibered dimension functiond: given a modelMofTand a definable setX⊆Mn, there is a definable typepinX, definable over a code forXand of the samed-dimension asX. Both o-minimal theories and the theory of closed ordered differential fields are shown to have this property. As an application, we derive a new proof of elimination of imaginaries for CODF.

We prove some results about the model theory of fields with a derivation of the Frobenius map, especially that the model companion of this theory is axiomatizable by axioms used by Wood in the case of the theory $\operatorname {DCF}_p$ and that it eliminates quantifiers after adding the inverse of the Frobenius map to the language. This strengthens the results from [4]. As a by-product, we get a new geometric axiomatization of this model companion. Along the way we also prove (...) a quantifier elimination result, which holds in a much more general context and we suggest a way of giving “one-dimensional” axiomatizations for model companions of some theories of fields with operators. (shrink)

Let T be an algebraically bounded theory. We consider the L ( δ ¯ ) $L(\bar \delta )$ upper L left parenthesis delta overbar right parenthesis -expansions of T by a tuple δ ¯ $\bar {\delta }$ delta overbar of derivations (which may be commuting or not). We investigate the model completion of either of the above theories, whose existence has been established in [21], with particular attention to its model-theoretic properties, including ω $\omega $ omega -stability, simplicity, open core, (...) and elimination of imaginaries. (shrink)

Pre-H-fields are ordered valued differential fields satisfying some basic axioms coming from transseries and Hardy fields. We study pre-H-fields that are differential-Hensel–Liouville closed, that is, differential-henselian, real closed, and closed under exponential integration, establishing an Ax–Kochen/Ershov theorem for such structures: the theory of a differential-Hensel–Liouville closed pre-H-field is determined by the theory of its ordered differential residue field; this result fails if the assumption of closure under exponential integration is dropped. In a two-sorted setting with one sort for a differential-Hensel–Liouville (...) closed pre-H-field and one sort for its ordered differential residue field, we eliminate quantifiers from the pre-H-field sort, from which we deduce that the ordered differential residue field is purely stably embedded and if it has NIP, then so does the two-sorted structure. Similarly, the one-sorted theory of differential-Hensel–Liouville closed pre-H-fields with closed ordered differential residue field has quantifier elimination, is the model completion of the theory of pre-H-fields with gap $0$, and is complete, distal, and locally o-minimal. (shrink)

We generalise the exponential Ax–Schanuel theorem to arbitrary linear differential equations with constant coefficients. Using the analysis of the exponential differential equation by Kirby :445–486, 2009) and Crampin we give a complete axiomatisation of the first order theories of linear differential equations and show that the generalised Ax–Schanuel inequalities are adequate for them.

We study definable types in the theory of closed ordered differential fields. We show a condition for a type to be definable, then we prove that definable types are dense in the Stone space of CODF.

We investigate the existence of “generic derivations” in exponential fields. We show that exponential fields without additional compatibility conditions between derivation and exponentiation cannot support a generic derivation.

We use the model completeness and axiomatisation of the theory of closed ordered differential fields to give a differential version of Dubois, Krivine and Risler's nullstellensatz and Stengle's positivstellensatz for ordered fields.

This article describes the emergence of formal methods in theory of partial differential equations in the French school of mathematics through Janet’s work in the period 1913–1930. In his thesis and in a series of articles published during this period, Janet introduced an original formal approach to deal with the solvability of the problem of initial conditions for finite linear PDE systems. His constructions implicitly used an interpretation of a monomial PDE system as a generating family of a multiplicative set (...) of monomials. He introduced an algorithmic method on multiplicative sets to compute compatibility conditions, and to study the problem of the existence and the uniqueness of a solution to a linear PDE system with given initial conditions. The compatibility conditions are formulated using a refinement of the division operation on monomials defined with respect to a partition of the set of variables into multiplicative and non-multiplicative variables. Janet was a pioneer in the development of these algorithmic methods, and the completion procedure that he introduced on polynomials was the first one in a long and rich series of works on completion methods which appeared independently throughout the twentieth-century in various algebraic contexts. (shrink)

In a recent article, Freitag, Moosa, and the author showed that in differentially closed fields of characteristic zero, if two types are nonorthogonal, then their n+3 and m+3 Morley powers are not weakly orthogonal, where n and m are their respective Lascar ranks. In this short note, we prove that the bound is tight: There are such types with weakly orthogonal n+2 and m+2 Morley powers. The types in question were constructed by Freitag and Moosa as examples of types with (...) degree of nonminimality 2. As interesting as our result are our methods: We rely mostly on Galois theory and some descent argument for types, combined with the failure of the inverse Galois problem over constant parameters. (shrink)